# The Embedding Flow of 3-Dimensional Locally Hyperbolic \(C^\infty \) Diffeomorphisms

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## Abstract

For finite dimensional locally hyperbolic \(C^\infty \) smooth real diffeomorphisms, the problem on the existence and smoothness of their embedding flows was solved by Li et al. (Am J Math 124:107–127, 2002) in the weakly non-resonant case. Whereas there are no general results on the existence of embedding flows in weakly resonant cases, except for some examples and some ones on the non-existence of embedding flows. The simplest hyperbolic diffeomorphisms having weakly resonant phenomena should be the 3-dimensional ones. This paper will concentrate on the embedding flow problem for this kind of diffeomorphisms. In one case we obtain complete characterization of the diffeomorphisms which admit embedding flows, and in the other cases we obtain some necessary conditions for the diffeomorphisms to have embedding flows. Also in this latter case we provide a sufficient condition for the existence of \(C^\infty \) embedding flow. As we know it is the first general result on the existence of embedding flows for locally hyperbolic \(C^\infty \) diffeomorphisms with weak resonance. These results show that the embedding flow problem in the weakly resonant case is much more involved.

## Keywords

3-Dimensional local diffeomorphism Weak resonance Hyperbolicity \(C^\infty \) smoothness Embedding flow## Mathematics Subject Classification

34A34 34C20 34C41 37G05## Notes

### Acknowledgments

The author is partially supported by NNSF of China grants 10831003 and 11271252, RFDP of Higher Education of China grant 20110073110054 and FP7-PEOPLE-2012-IRSES-316338 of Europe, and by innovation program of Shanghai municipal education commission grant 15ZZ012.

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